${\mathcal{H}}_2$ static and dynamic output-feedback control problems are investigated for linear time-invariant uncertain systems. The goal is to minimize the averaged ${\mathcal{H}}_2$ performance in the presence of nonlinear dependence on time-invariant probabilistic parametric uncertainties. By applying the polynomial chaos theory, the control synthesis problem is solved using a high-dimensional expanded system which characterizes stochastic state uncertainty propagation. Compared to existing polynomial chaos-based control designs, the proposed approach addresses the simultaneous presence of parametric uncertainties and white noises. The effect of truncation errors due to using finite-degree polynomial chaos expansions is captured by time-varying norm-bounded uncertainties, and is explicitly taken into account by adopting a guaranteed cost control approach. This feature avoids the use of high-degree polynomial chaos expansions to alleviate the destabilizing effect of expansion truncation errors, thus significantly reducing computational complexity. Moreover, rigorous analysis clarifies the condition under which the stability of the high-dimensional expanded system implies the internal mean square stability of the original system under control. Using iterations between synthesis and post-analysis, a bisection algorithm is proposed to find the smallest bounding parameter on the effect of expansion truncation errors such that robust closed-loop stability is guaranteed. A numerical example is used to illustrate the effectiveness of the proposed approach.

${\mathcal{H}}_2$研究了线性时不变不确定系统的静态和动态输出反馈控制问题。目标是最小化平均${\mathcal{H}}_2$ 在对时不变概率参数不确定性存在非线性依赖性的情况下的性能. 通过应用多项式混沌理论，使用表征随机状态不确定性传播的高维扩展系统来解决控制综合问题。与现有的基于多项式混沌的控制设计相比，所提出的方法解决了参数不确定性和白噪声的同时存在问题。由于使用有限次多项式混沌展开而导致截断误差的影响由时变范数有界不确定性捕获，并通过采用有保证的成本控制方法明确考虑在内。该特性避免了使用高次多项式混沌展开来减轻展开截断误差的不稳定效应，从而显着降低计算复杂度。此外，严谨的分析阐明了高维扩展系统的稳定性意味着受控原始系统的内部均方稳定性的条件。利用综合和后分析之间的迭代，提出了一种二分算法来寻找对扩展截断误差影响的最小边界参数，从而保证稳健的闭环稳定性。一个数值例子被用来说明所提出方法的有效性。